Predstavitev polinomskih krivulj in ploskev s principom razcveta

magistrsko delo

Matija Šteblaj (Author), Jan Grošelj (Mentor)

Abstract

V delu predstavimo princip razcveta: za vsako polinomsko preslikavo F stopnje n med afinimi prostori obstaja simetrična, multiafina preslikava f v n spremenljivkah, ki se na diagonali ujema s prvotno preslikavo. Preslikavi f rečemo (multiafin) razcvet preslikave F. Z njo lahko vsako polinomsko krivuljo oz. ploskev predstavimo kot Bezierjevo krivuljo oz. trikotno Bezierjevo krpo. Pri tem so kontrolne točke krivulje oz. krpe določene z vrednostmi razcveta na izbranem afinem ogrodju prostora. S pomočjo razcveta obravnavamo tudi zlepke Bezierjevih krivulj oz. krp in njihovo gladkost. Pokažemo, da se pogoji za posamezen red gladkosti lahko izrazijo kot enakosti med razcveti obeh krivulj oz. krp pri ustreznih argumentih.

Keywords

matematika;afini prostori;polinomi;razcvet;Bernsteinovi bazni polinomi;Bézierjeve krivulje;trikotne Bézirjeve krpe;ploskve;de Casteljaujev algoritem;zlepki;

Data

Language:	Slovenian
Year of publishing:	2023
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[M. Šteblaj]
UDC:	519.6
COBISS:	138055171
Views:	974
Downloads:	88
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Representation of polynomial curves and surfaces with the blossoming principle
Secondary abstract:	In this work we present the blossoming principle: for every polynomial map F of degree n between affine spaces there exists a symmetric, multiaffine map f in n variables, which agrees with F on the diagonal. We call f the (multiaffine) blossom of F. With it we can represent each polynomial curve or surface as a Bezier curve or triangular Bezier patch. The control points of said curve or patch are determined by the values of the blossom f on a chosen affine frame. Utilising the blossoming principle, we also describe splines of Bezier curves and splines of triangular Bezier patches and their smoothness. We show that the conditions for a spline to satisfy a particular order of continuity can be expressed as equalities between the blossoms of both curves or surfaces on a specific collection of arguments.
Secondary keywords:	mathematics;affine spaces;polynomials;blossom;Bernstein basis polynomials;Bézier curves;triangular Bézier patch;surfaces;de Casteljau algorithm;splines;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja; Pedagoška matematika
Pages:	IX, 67 str.
ID:	17798987